Continuous maps of the interval with finite nonwandering set
HTML articles powered by AMS MathViewer
- by Louis Block PDF
- Trans. Amer. Math. Soc. 240 (1978), 221-230 Request permission
Abstract:
Let f be a continuous map of a closed interval into itself, and let $\Omega (f)$ denote the nonwandering set of f. It is shown that if $\Omega (f)$ is finite, then $\Omega (f)$ is the set of periodic points of f. Also, an example is given of a continuous map g, of a compact, connected, metrizable, one-dimensional space, for which $\Omega (g)$ consists of exactly two points, one of which is not periodic.References
- Louis Block, Diffeomorphisms obtained from endomorphisms, Trans. Amer. Math. Soc. 214 (1975), 403–413. MR 388457, DOI 10.1090/S0002-9947-1975-0388457-6
- Louis Block, Morse-Smale endomorphisms of the circle, Proc. Amer. Math. Soc. 48 (1975), 457–463. MR 413186, DOI 10.1090/S0002-9939-1975-0413186-5
- Louis Block, The periodic points of Morse-Smale endomorphisms of the circle, Trans. Amer. Math. Soc. 226 (1977), 77–88. MR 436220, DOI 10.1090/S0002-9947-1977-0436220-1
- Rufus Bowen and John Franks, The periodic points of maps of the disk and the interval, Topology 15 (1976), no. 4, 337–342. MR 431282, DOI 10.1016/0040-9383(76)90026-4
- Michael Shub, Endomorphisms of compact differentiable manifolds, Amer. J. Math. 91 (1969), 175–199. MR 240824, DOI 10.2307/2373276
- S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747–817. MR 228014, DOI 10.1090/S0002-9904-1967-11798-1
- O. M. Šarkovs′kiĭ, Co-existence of cycles of a continuous mapping of the line into itself, Ukrain. Mat. . 16 (1964), 61–71 (Russian, with English summary). MR 0159905
Additional Information
- © Copyright 1978 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 240 (1978), 221-230
- MSC: Primary 54H20; Secondary 58F20
- DOI: https://doi.org/10.1090/S0002-9947-1978-0474240-2
- MathSciNet review: 0474240